Logarithm
Here is the plot of the complex logarithm function:
\(f(z) = \log\,z\;\textrm{(principal branch)},\quad z \in \mathbb{C}(0,3)\)
We may observe that it has a cut, also along the negative half of the real axis.
Furthermore this function is another good example for what we have stated in the case of the sine function. One could recognise the real logarithm function on the picture. (The positive real numbers give its domain, near 0 function values tending to minus infinity are found, it has a zero at 1, and—though quite slowly—tends to plus infinity. In colors: white, cyan, black, red, white.)
The logarithm is the inverse of the exponential function which is not injective (but \(2 \pi i\) periodic), so it is to be restricted to some 'horizontal stripe' with the width of \(2 \pi\). (See figure below.) The so-called principal branch is the result of the 'natural' choice of the bounds: \(-\pi\) and \(\pi\). (A matter of convention.) We can obtain the side branches by choosing other 'stripes'. Of course the choice of the domain also influences the location of the branch cut.
We also included the picture of the exponential function and the coloring for assistance:
A bunch of branches of the logarithm function are illustrated in the video below, which could also be downloaded here.
In this animation the domain of restriction is moved up and down in a sinusoidal manner with an amplitude of \(\pi\) in both directions. (The cut adapts its angle accordingly.)